This invention relates generally to apparatus and methods for identifying lattice structures, and more particularly to apparatus and methods for identifying intralattice and interlattice relationships.
A knowledge of the nature of the lattice structure of a material, as well as how it relates to the lattice structures of other materials, is essential in any systematic analysis of physical properties, and has many important commercial applications. For example, in crystallography, it is important for further analysis and identification of a crystalline structure under investigation to determine its symmetry characteristics. As another example, in materials design, once a new compound has been synthesized with a sought-after property, it can be extremely useful to the researcher to find all other materials that bear some specified lattice relationship. Thus, for example, once related compounds have been identified which bear a given lattice relationship to a new superconducting material, the related compounds can then be evaluated to see if they also exhibit superconductivity.
As still other examples, it can be important for the researcher to know whether two apparently different materials exhibiting the same property have the same or a derivative lattice relationship, or to identify an unknown phase by matching the unknown against all known lattice structures. Additionally, the researcher may wish to analyze structural lattice relationships within a large set of compounds, or between two sets of compounds.
Heretofore, analysis of lattice structures, both to determine intralattice relationships (e.g., lattice symmetries) and to determine interlattice relationships, has been difficult, cumbersome and subject to substantial error. For example, in the collection of crystallographic data, the experimentalist traditionally has relied on familiar or standard orientations to guide both the initial collection of data and evaluation thereof to define the lattice and the crystal symmetries. On a diffractometer, for example, a conventional unit cell, as defined by the magnitudes of the cell parameters, is determined and the assumed Laue symmetry is verified by taking specially oriented films or by checking the intensities of equivalent (h,k,l)'s listed for standard orientations. While there are many valid reasons for choosing conventional cells and orientations in the latter stages of experimental work, by choosing specific or familiar orientations in the initial stages, assumptions are made which influence what data are collected, and consequently, mistakes are more likely to be made.
In accordance with the present invention, a converse transformation matrix generation approach is used either i) to relate a lattice structure of one material to the lattice structure(s) of one or more other materials for determining interlattice relationships which allow materials to be identified and classified relative to other materials; or ii) to relate a lattice structure of a material to itself for determining lattice symmetry. The matrix approach is an extremely powerful, efficient and flexible analytical tool which is readily implemented and avoids the constraints and error inherent in the conventional approaches heretofore used. For example, the matrix approach is substantially more effective than prior approaches because it will maintain its selectivity in matching lattice structures despite the rather large experimental errors that are routinely associated with electron diffraction data. As another example, the determination of lattice symmetry using the matrix approach of the present invention does not require that the lattice and its symmetry be expressed with respect to a standard cell or a standard orientation. The properties of the lattice are reflected in any primitive cell because translation of the primitive unit cell generates the entire lattice. In accordance with the present invention, the symmetry matrices which transform the lattice into itself are generated, and are used to determine the metric symmetry and any pseudosymmetry, as well as the Laue symmetry, group-subgroup relationships, the nature and directions of symmetry axes, and conventional or standard cells. In addition, the determination of standard or conventional cells is greatly simplified. In contrast to other methods for determining standard cells, the matrix approach of the present invention permits working with symmetry directly in the form of matrices, and not with the magnitude of lattice parameters and their associated errors. Calculations are straightforward and a transformation matrix is found using linear algebra techniques.
Still further, the matrix approach of the present invention enables computer-based controllers for commercial diffractometers (x-ray, neutron and electron) to be implemented which fully automate the diffractometry process in a theoretically and experimentally correct and error-free manner. Unlike procedures currently used in diffractometry, errors in strategy are impossible with the matrix approach because at each step exactly the right data for control decisions are directly available in a clear, logical and concise format. Since the matrix approach is extremely reliable, its use will prevent errors in symmetry and structure determinations, which is widely recognized as a very serious current problem resulting in erroneous symmetry determinations in about five percent of the approximately 25,000 full structure determinations carried out annually.
The matrix approach of the present invention also enables electron diffractometry to be converted from a two dimensional technique that focuses primarily on d-spacings to a three dimensional technique similar to that employed in single crystal x-ray and neutron diffractometry. The ability to determine the cell structure and symmetry from data collected on extremely small samples and to identify the structure using computerized databases of known structures represents a new and comprehensive method to identify crystalline phases.
The matrix approach of the present invention thus represents a powerful new strategy for lattice structure analysis in which the emphasis is shifted from standard cells and standard orientations to matrices.
A portion of the present invention's matrix approach to lattice analysis in the context of determining symmetry has been discussed in detail by the present applicants in their article "A Matrix Approach to Symmetry", Acta Cryst. (1987) A43, pp. 375-384, which is hereby incorporated by reference. However, no method for generating the necessary symmetry matrices which relate any observed primitive cell of a lattice to itself is described in the aforesaid article, or in any other published article. The general converse transformation method described herein which applicants have developed generates matrices relating any two lattice cells (including a lattice cell to itself); and can be used in new, more efficacious methods for control of diffractometers, and for phase identification using analytical electron microscope (AEM) data and existing large scale databases providing chemical, physical and crystallographic data on solid-state materials.
Accordingly, it is a primary object of the present invention to provide apparatus and methods for identifying lattice structures using the converse transformation matrix generation method of the present invention.
It is a further primary object of the present invention to provide apparatus and methods for identifying unknown materials using electron diffraction and energy dispersive spectroscopy data, databases on solid-state materials and the converse transformation matrix generation method of the present invention.
It is a still further primary object of the present invention to provide improved diffractometry apparatus and methods using the converse transformation matrix generation method of the present invention.
In accordance with the present invention, a method of comparing two crystalline materials to determine whether they have a predetermined lattice structure relationship therebetween comprises the steps of:
a) determining primitive lattice cells Y and Z, respectively, for the two materials, the cells Y and Z having three cell edges YA, YB, YC and ZA, ZB and ZC, respectively, and three cell angles YAL, YBE, YGA and ZAL, ZBE and ZGA, respectively;
b) generating all matrices H, if any, which transform cell Z into cell Y within predetermined maximum cell edge and angle tolerances TOLI1, TOLI2, TOLI3, and TOLI4, TOLI5, TOLI6, respectively; and if at least one matrix H is generated;
c) analyzing the nature of the generated matrix (matrices) H and its inverse (their respective inverses) H' to determine the nature of the lattice relationship:
1) If a matrix H has integer matrix elements and a determinant HDET=1, then Cell Z and Cell Y define the same lattice;
2) If a matrix H or its inverse H' has integer matrix elements and a determinant HDET greater than one, then Cell Z and Cell Y are in a subcell/supercell relationship; or
3) If a matrix H and its inverse H' both have one or more fractional matrix elements, then Cell Z and Cell Y define lattices that are in a composite relationship.
In accordance with a further aspect of the present invention, a method for analyzing the symmetry of a crystalline material comprises the steps of:
a) collecting edge data ZA, ZB and ZC and angle data ZAL, ZBE and ZGA defining any primitive lattice cell Z of the material;
b) generating all symmetry matrices Hs which transform cell Z into itself within predetermined maximum cell edge and angle parameter tolerances TOLI1, TOLI2, TOLI3 and TOLI4, TOLI5, TOLI6, respectively;
c) determining the metric symmetry using symmetry matrices Hs; and
d) defining the crystal symmetry using symmetry matrices Hs.
In accordance with a still further aspect of the present invention, a method for identifying an unknown crystalline material comprises the steps of;
a) determining a primitive lattice cell Z of the unknown material, the cell Z having three cell edges ZA, ZB, and ZC, respectively, and three cell angles ZAL, ZBE, and ZGA, respectively;
b) determining the chemical composition of the unknown material;
c) searching a database comprising lattice cell data and element type data for materials with known lattice structures and chemical compositions by at least in part generating matrices H identifying all compounds having lattice cell structures related to cell Z;
d) analyzing the matrices H to identify which of the compounds identified in step c) match cell Z by having a lattice cell structure identical to or in a subcell/supercell derivative relationship to cell Z, and saving the lattice cell matching compounds as a first data set;
e) searching the database for all compounds which match the unknown material by having the same element types as the unknown material, and saving the element type matching compounds as a second data set; and
f) combining the first and second data sets to derive all known compounds having the same lattice cell structure and element types.
In accordance with another aspect of the present invention, in each of the foregoing methods, the matrices H are generated by a converse transformation method comprising the steps of:
finding all matrix triples AU, AV, AW; BU, BV, BW; CU, CV, CW which accomplish transformation of the respective Z-cell edges to the corresponding edges of the desired cell within the corresponding ones of the maximum cell edge tolerances; and
finding all combinations of the matrix triples found in the matrix-triple-finding step which accomplish transformation of the respective Z-cell angles to the corresponding angles of the desired cell within the corresponding ones of the maximum acceptable cell angle tolerances.
These and other objects, features and advantages of the present invention will be described in or apparent from the following detailed description of preferred embodiments.